Perhaps the best thing to do, before giving some examples on the correct way to solve an empowerment operation in which the number that serves as a base is a decimal number, is to revise briefly the definition of this operation, in order to understand each one of the exercises that are presented in its just mathematical context.
Decimal number powers
In this sense, we can begin by saying that the Potentiation of decimal numbers is that operation destined to discover which is the product obtained by taking the decimal number that serves as base, and multiplying it by itself, as many times as the integer that acts as exponent indicates. Hence, some authors point out that the Potentiation of decimal numbers can also be interpreted as an abbreviated sum of these.
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Steps to solve a decimal number potency
However, precisely because of the nature of the number that serves as the base, i.e. the decimal number, which is composed of an integer and a decimal part, it is necessary to follow a different method to that which is followed when the base is integer, in order to avoid making mistakes, when the exercise must be solved manually.
Consequently, the steps that must be followed when solving any operation of powers of decimal numbers will be the following ones:
- Once the potentiation operation has been proposed, each one of the elements must be reviewed in order to know the nature of each one of them.
- Once it has been found that the operation is established between a decimal number and a positive integer exponent, the decimal number comma should be deleted so that it performs as an integer for the duration of the operation.
- Once this has been done, the number from which the comma has been removed is then elevated to the indicated exponent.
- Once the obtained power has been obtained, a second operation must be carried out to determine the specific place where the comma of the final result must be located, since the power of a decimal number must essentially give a decimal number. In this way, we will proceed to count how many incomplete units the decimal number had at the time of starting the operation. This figure should be multiplied by the value of the exponent. The result of this multiplication will be the number of spaces that must be counted from right to left, in the power obtained, before placing the comma.
- Therefore, with the result of the multiplication of the number of incomplete units and the value of the exponent, the comma is placed at the power obtained, assuming the decimal number obtained as the final result of the operation, that is, as the power of the decimal number that was originally set as the base.
- If you want to check this operation, you will see the inverse operation, that is to say, the root. Consequently, the power obtained will be taken as radicando, the exponent as index, and the result of this other operation should give the number that has served as base for the potentiation. If so, the operation has been resolved correctly.
Examples of powers of decimal numbers
However, perhaps the best way to complete an explanation about the correct way in which any power of decimal numbers must be resolved is through a series of examples that allow us to see in a concrete way how each one of the steps indicated by Mathematics are fulfilled. Here are some of them:
Solve the following decimal number power operation: 2.53 =
As the method suggested by the mathematical discipline for solving this exercise points out, one should begin by suppressing the comma of the decimal that serves as the base, and then elevate it to exponent 3 to which it is elevated:
2,53 → 253 = 15625
Once this result is found, it will be time to locate the comma again. To do this, multiply the value of the exponent by the number of incomplete units or decimal parts that the number that served as base had, being in this case equal to 1:
3 x 1 = 3
The product obtained shall be the number of spaces to be counted from right to left before placing the comma on the power:
15625 → 15,625
Having done this, one can then consider the operation resolved. Therefore, the only thing left to do is to express the result obtained:
2,53 = 15,625
Resolve the following decimal number power operation:
In this case, once the numbers on which the potentiation operation has been raised have been reviewed, it can be found that it is not only a decimal base, but that the whole part of it is equal to zero. However, this should not represent a big problem when solving the operation, since when the comma is suppressed, simply the left zero will not have any value, so it will not be necessary to take it into account when solving the power:
0,444 → 444
When the operation is expressed as an integer, the proposed power is resolved:
444 = 3748096
We then look for where the comma should be located, multiplying the number of decimals that had the original base and the value of the exponent:
2 x 4= 8
This product is the number of spaces that will be counted from right to left before the comma is placed:
3748096 → 0,03748096
The quantity of elements of the power obtained was less than the product obtained between the number of decimals of the base and the value of the exponent, so it was necessary then to add zeros in the power, when locating the comma. Once this number is obtained, the final result of the operation will be expressed:
0,444 = 0,03748096
Solve the following power: 1,2342 =
In this case, we will also start by removing the comma from the decimal power, in order to take the number as an integer, and be able to raise it to the square:
1,2342 → 12342 = 1522756
The number of decimals or incomplete units of the original base is then multiplied by the value of the exponent:
3 x 2= 6
This product will be translated in the number of places that will have to be counted from the right to the left, to be able to locate the comma in the obtained power:
1522756 → 1,522756
Once this decimal number is obtained, considered the final result, the next step will consist of expressing the operation as resolved:
1,2342 = 1,522756